In this thesis, a series of mathematical models suitable for describing biological tissue growth are developed. The motivation for this work is a bioreactor system which provides perfusion and compressive mechanical stimulation to a cell-seeded scaffold; however, the formulation is sufficiently general to be applied to a vast range of tissue engineering applications. Our models are used to investigate the influence of (i) cell-cell and cell-scaffold interactions, and (ii) the mechanical environment, on tissue growth.
In the first part of the thesis, we extend a model due to Franks (2002) (in which the cell and culture medium phases are represented by viscous fluids) by including perfusion and coupling the cells' response to their environment. Specifically, we consider the effect of the cell density and pressure on tissue growth. We analyse the model using analytic and numerical techniques; numerical simulations suggest that comparison of construct morphology in the presence and absence of perfusion provides a means to identify the dominant regulatory growth stimulus.
The solid characteristics of the construct and interactions between the cells and scaffold are necessarily neglected in the two phase model. Guided by this, we develop more complex three phase models. Using numerical simulations, the influence of cell-cell and cell-scaffold interactions is investigated and less porous scaffolds are shown to improve control over cell behaviour. We use the model to compare the cells' response to different regulatory stimuli, including flow-induced shear stress. Our results suggest that uniform initial cell seeding and stimulating cell movement are crucial in maintaining the mechanical integrity of tissue constructs.
We also study the effect of scaffold compression on the mechanical environment of the cells contained within, developing both a classical Biot formulation and a multiphase model. We demonstrate that the bioreactor geometry introduces significant spatial variation in the mechanical stimuli relevant to tissue growth and that such considerations will play a key role in comprehensive models of mechanotransduction-affected growth.